Impossible to fix leaky faucet

People,

Popped in a faucet on a sink - faucet worked well before on another sink. Now, faucet has a steady drizzle of water. Cold and hot water handles are closed all the way, of course. Why would this faucet, which worked great on another sink drizzle steadily on this sink?

Is it the goose neck base seal or would you suspect the valve at the cold? (hot side is shut off/blocked, as we use cold only here- no hot water tank hooked up).